Fully Implicit Form of Differential Quadrature Method for Multi-Species Solute Transport in Porous Media*
Fully Implicit Form of Differential Quadrature Method for Multi-Species Solute Transport in Porous Media*
Solute transport problems, including sequential multi-species transport phenomena, frequently occur in soil systems. The goal of this paper is to present a novel one-dimensional numerical model with a fully implicit form of differential quadrature method for solving multi-species solute transport equations. The analytical results of three multi-species solute dispersion problems with three- and four-chain members are used to analyse the developed model. Simultaneously, the outcomes of the developed model are compared with the performance of the fully implicit fourth-order finite difference method. Finally, the accuracy of the established model is discussed and evaluated. According to the numerical experiments, the derived model is very useful and widely applicable.
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- [1] Bagalkot, N., Kumar, G.S., 2015. Effect of nonlinear sorption on multispecies radionuclide transport in a coupled fracture-matrix system with variable fracture aperture: a numerical study. ISH Journal of Hydraulic Engineering. 21(3) 242–254, http://dx.doi.org/10.1080/09715010.2015.1016125
- [2] Bauer, P., Attinger, S., Kinzelbach, W., 2001. Transport of a decay chain in homogenous porous media: analytical solutions, Journal of Contaminant Hydrology. 49, 217–239.
- [3] Bellman, R., Casti, J., 1971. Differential quadrature and long-term integration. J. Math. Anal. Appl. 34 (2), 235–238.
- [4] Chaudhary, M., Singh, M.K. 2020, Study of multispecies convection-dispersion transport equation with variable parameters, Journal of Hydrology 591 (2020), https://doi.org/10.1016/j.jhydrol.2020.125562
- [5] Chen, W. and Zhong, T., 1997, The study on the nonlinear computations of the DQ and DC methods. Numerical Methods for Partial Differential Equations, 13, 57–75
- [6] Chen-Charpentier, B.M., Dimitrov, D.T., Kojouharov H.V., 2009. Numerical simulation of multi-species biofilms in porous media for different kinetics, Mathematics and Computers in Simulation, 79, 1846–1861.
- [7] Ciftci, E., 2017. Modelling coupled density-dependent flow and solute transport with the differential quadrature method, Geosciences Journal, 21(5), 807817, http://dx.doi.org/10.1007/s12303-017-0009-5
- [8] Konikow, L.F., and Bredehoeft, J.D., 1978, Computer model of two-dimensional solute transport and dispersion in ground water: U.S. Geological Survey, Techniques of Water-Resources Investigations, Book 7, Chap. C2.
- [9] Civalek, O. 2004, Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns, Engineering Structures, 26,171–186.
- [10] Das, P., Akhter, A., Singh, M.K., 2018. Solute transport modelling with the variable temporally dependent boundary, Sadhana, 43, 1-12, https://doi.org/10.1007/s12046- 017-0778-6
- [11] Essaid, H.I. and Bekins, B.A., 1997, BIOMOC, A Multispecies Solute-Transport Model with Biodegradation, U.S. GEOLOGICAL SURVEY, Water-Resources Investigations Report 97-4022.
- [12] Gharehbaghi, A., 2016. Explicit and implicit forms of differential quadrature method for advection–diffusion equation with variable coeffcients in semi-infnite domain, J. Hydrol. 541 (1), 935–940, http://dx.doi.org/10.1016/j.jhydrol.2016.08.002
- [13] Gharehbaghi, A., 2017. Third- and fifth-order finite volume schemes for advection– diffusion equation with variable coefficients in semi-infinite domain. Water and Environment Journal. 31 (2), 184–193. doi:10.1111/wej.12233
- [14] Gharehbaghi, A., Kaya, B., Saadatnejadgharahassanlou, H., 2017. Two-dimensional bed variation models under nonequilibrium conditions in turbulent streams. Arabian Journal for Science and Engineering. 42 (3), 999–1011.
- [15] Kaya, B., 2010. Solution of the advection-diffusion equation using the differential quadrature method. KSCE J. Civil Eng. 14 (1), 69–75.
- [16] Kaya, B., Arisoy, Y., 2011. Differential quadrature solution for one-dimensional aquifer flow, Mathematical and Computational Applications, 16(2), 524-534.
- [17] Kaya, B., Gharehbaghi., A., 2014. Implicit Solutions of Advection Diffusion Equation by Various Numerical Methods. Aust. J. Basic & Appl. Sci., 8(1): 381-391.
- [18] Kumar, A., Kumar, D., Kumar, J.N., 2010. Analytical solutions to one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media. J. Hydrol. 380, 330–337.
- [19] Massoudieh, A., Ginn, T.R., 2007, Modeling colloid-facilitated transport of multispecies contaminants in unsaturated porous media, Journal of Contaminant Hydrology 92, 162–183, doi:10.1016/j.jconhyd.2007.01.005
- [20] Engdahl N.B., Aquino, T., 2018, Considering the utility of backward-in-time simulations of multi-component reactive transport in porous media, Advances in Water Resources, 119, 17–27. https://doi.org/10.1016/j.advwatres.2018.06.003
- [21] Natarajan N., Kumar S.G., 2010. Finite difference approach for modeling multispecies transport in porous media, International Journal of Engineering Science and Technology. 2(8), 3344-3350.
- [22] Pathania, T., Eldho, T.I., Bottacin-Busolin, A. 2020. Coupled simulation of groundwater flow and multispecies reactive transport in an unconfined aquifer using the element-free Galerkin method. Engineering Analysis with Boundary Elements. 121, 31–49.
- [23] Pérez Guerrero, J.S., Skaggs, T.H., van Genuchten, M.Th., 2009. Analytical Solution for Multi-Species Contaminant Transport Subject to Sequential First-Order Decay Reactions in Finite Media. Transp Porous Med. 80, 373–387, DOI 10.1007/s11242- 009-9368-3
- [24] Ramos, T.B., Šimůnek, J., Gonçalves, M.C., Martins, J.C., Prazeres, A., Castanheira, N.L., Pereira, L.S., 2011, Field evaluation of a multicomponent solute transport model in soils irrigated with saline waters, Journal of Hydrology, 407(1–4), 129-144, https://doi.org/10.1016/j.jhydrol.2011.07.016.
- [25] Savovic, S., Djordjevich, A., 2012. Finite difference solution of the one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media. Int. J. Heat Mass Transf. 55, 4291–4294. http://dx.doi.org/10.1016/j. ijheatmasstransfer.2012.03.073
- [26] Sharma A., Guleria, A., Swami D., 2016. Numerical modelling of multispecies solute transport in porous media. Hydro 2016 international conference. 159-169.